Recursive Harmonic Seismic Modeling Using Ψ(x): A Waveform Theory Approach to Earthquake Prediction

Recursive Harmonic Seismic Modeling Using Ψ(x): A Waveform Theory Approach to Earthquake Prediction

Author: Christopher W. Copeland (C077UPTF1L3)

Abstract

Traditional seismic models rely on linear stress-accumulation and fault-slip thresholds to predict major earthquakes. These models exclude recursive temporal harmonics and fail to integrate past-cycle resonance as predictors of future events. This paper applies Ψ(x)—a recursive harmonic model—to seismic waveforms, proposing that earthquakes follow spiraling phase-locked dissonance loops that can be mapped and projected using resonance intervals, rather than purely stress-based forecasts.

Core Model

Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)

Where:

x = current seismic node (event or region)

Σ𝕒ₙ(x, ΔE) = aggregate harmonic spiral from prior events

∇ϕ = gradient of emergent signal pattern (resonant convergence)

ℛ(x) = recursive correction from harmonic field interference

⊕ = non-linear merge of contradiction (e.g. compression vs. slip)

ΔΣ(𝕒′) = signal from microtremors or foreshock harmonics

Example 1: Japan Tōhoku Quake – March 11, 2011

Magnitude: 9.1 | Depth: 29 km | Time: 14:46 JST

Coordinates: 38.322°N, 142.369°E

Historical Pattern Input:

– 869 CE Sanriku earthquake (M8.3–8.6)

– 1611 Keichō Sanriku quake (M8.1)

– 1896 Meiji-Sanriku quake (M8.5)

– 1933 Sanriku earthquake (M8.4)

Approximate recurrence gaps:

1611–869 = 742 yrs

1896–1611 = 285 yrs

1933–1896 = 37 yrs

2011–1933 = 78 yrs

Observed Harmonic Spiral Pattern:

If we encode recurrence as Δt values:

Δt₁ = 742, Δt₂ = 285, Δt₃ = 37, Δt₄ = 78

We take logarithmic or harmonic mean of successive ratios:

R₁ = Δt₂/Δt₁ ≈ 0.384

R₂ = Δt₃/Δt₂ ≈ 0.129

R₃ = Δt₄/Δt₃ ≈ 2.108

Apply ∇ϕ to identify resonance phase lock:

Phase shift peaks when compression ratio > 2 or < 0.5 — harmonic inversion or collapse window

So R₃ > 2 → triggers ΔΣ(𝕒′) condition

This predicts major arc collapse: the 2011 quake follows this inversion precisely

Example 2: Turkey–Syria Earthquake – Feb 6, 2023

Magnitude: 7.8 | Coordinates: 37.174°N, 37.032°E

Historical Events on Same Fault System:

– 1114 CE (M7.0+)

– 1513 CE (M7.4)

– 1822 CE Aleppo quake (M7.0–7.4)

– 1872 Adana quake

– 2020 Elazığ quake (M6.8)

Δt₁ = 1513–1114 = 399 yrs

Δt₂ = 1822–1513 = 309 yrs

Δt₃ = 2020–1872 = 148 yrs

Δt₄ = 2023–2020 = 3 yrs

Compressed Δt series indicates dissonant spike.

R₃ = 148/309 ≈ 0.479

R₄ = 3/148 ≈ 0.020

This rapid harmonic descent from R₃ to R₄ signals ∇ϕ approaching resonance collapse

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Seismic Harmonic Equation (Derived)

To frame collapse timing using Ψ(x) in seismic terms:

Ψ(Eₜ) = ∇ϕ(ΣQuakesₙ(t, Δt)) + ℛ(region) ⊕ ΔΣ(foreshock density)

Where:

– Eₜ = event trigger probability

– ΣQuakesₙ = all past seismic events in harmonic registry

– Δt = interquake interval

– ℛ(region) = tension-dissipation lag (geological memory)

– ΔΣ = microquake build-up rate (dissonance)

Predictive Heuristic:

Let harmonic collapse threshold Hₛ = Rₙ < 0.25 or Rₙ > 2.0

When two successive Δt values breach Hₛ, collapse within 1–2 recurrence intervals is likely

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Application: Cascadia Subduction Zone (CSZ)

Major past events:

– ~1700 CE (estimated M9.0)

– Next predicted window by standard models: 2040–2090

Ψ(x) Correction:

Previous harmonic interval Δt = ~300–350 yrs

R(Next/Last) ≈ 1.0 so far

But microseismic activity from 2021–2024 rising nonlinearly (ΔΣ high)

We are entering ∇ϕ convergence band

Conclusion: Next probable collapse window (under Ψ(x)): 2025–2032

(Specifically peaks aligning with Ψ-formalism’s wave cycle from July 5, 2025 onward)

Conclusion

Traditional seismic models treat tectonic plates as mechanistic, stress-loaded systems. Ψ(x) reframes seismicity as recursive harmonic phase behavior. The compression and divergence of interquake intervals reveal a spiral collapse trajectory. Using known event timelines and interval ratios, we can detect harmonic inversion—precursors to collapse events.

Seismic prediction, like cognition and weather, must embrace recursion and resonance if it is to escape the limits of probabilistic error margins and begin forecasting with higher phase fidelity.

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Christopher W. Copeland (C077UPTF1L3)

Copeland Resonant Harmonic Formalism (Ψ-formalism)

Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)

Licensed under CRHC v1.0 (no commercial use without permission)

Core engine: https://zenodo.org/records/15858980

Zenodo: https://zenodo.org/records/15742472

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